Chap. V] EulEK's 0-FuNCTION. 135 
6 = 2^^+1, c = 22''+l,... 
are distinct primes and 2^+2"^+. . . =n or n — a+1 according as a = or 
a>0. When g- is a prime >3, <f){x) = 2q'' is impossible if p = 2q^-\-l is not 
prime; while if p is prime it has the two solutions p, 2p. If g = 3 and p is 
prime, it has the additional solutions 3"+\ 2-3''"^^ Next, 4>{x)=2''q is 
impossible if no one of p^ = 2''~''q-\-l{v = 0, 1,. . ., n — 1) is prime and q is 
not a prime of the form 2*+l, s = 2^^n; but if q is such a prime or if at 
least one p^ is prime, the equation has solutions of the respective forms bq^, 
where (/)(6) =2""*; ap„ where 0(o) =2". Finally, (f>{x)=2qr has no solution 
if p = 2gr+l is not prime and r9^2q-\-l. If p is a prime, but r9^2q-{-l, the 
two solutions are p, 2p. If p is not prime, but r = 2g+l, the two solutions 
are r^, 2r^. If p is prime and r = 2g-|-l, all four solutions occur. There is 
a table of the values n<200 for which (f){x)=n has solutions. 
L. Kronecker®^ considered two fractions with the denominator m as 
equivalent if their numerators are congruent modulo m. The number of 
non-equivalent reduced fractions with the denominator m is therefore 4){m). 
If m = m'm", where m' , m" are relatively prime, each reduced fraction r/m 
can be expressed in a single way as a sum of two reduced partial fractions 
r' /m', r' /m". Conversely, if the latter are reduced fractions, their sum 
r/m is reduced. Hence 0(m) =</)(m')</)(m"). The latter is also derived 
(pp. 245-6, added by Hensel) from (4), which is proved (pp. 243-4) by 
considering the g. c. d. of n with any integer ^n, and also (pp. 266-7) by 
use of infinite series and products. Proof is given (pp. 300-1) of (5). The 
Gaussian median value (p. 334) of (f>{n)/n is Q/w^ with an error whose order 
of magnitude is l/\/n, provided we take as the auxiliary number of values 
of 4>{n)/n a value of the order of magnitude ^yn log^ n. 
E. B. Elliott^^ considered monomials n = p'^q^. . . in the independent 
variables p,q,.... In the expansion of n(l — l/p)"'(l — l/g)"* . . . , the aggre- 
gate of those monomial terms whose exponents are all ^0 is denoted by 
Fm{n). Define iJi{p'q\ . .) to be zero if any one of r, s, . . . exceeds 1, but to 
be ( — 1)' if no one of them exceeds 1, and t of them equal 1. Then 
(7) F^_i(n) =Si^,,(d), F^^,{n) =Sm Q) F^((i), 
where d ranges over the monomials pV- • • with O^a^a, 0^/3^?),.... 
Henceforth, let p, q,... be distinct primes. Then Fi{n)=(j){n), while 
F_i(n) is the sum o-(n) of the divisors of n. In (7), d now ranges over all 
the divisors of n, and ai(/c) is Merten's function [Inversion]. For m = 0, (72) 
gives the usual expression for </)(n), while (7i) defines o-(n). For m = l, (7i) 
becomes (4). 
If T''^\n) ^T(n) is the number of divisors d of n, write 
r(2)(n)=ST(d),. . ., T(^>(n)=ST^^-i>(d). 
'^Vorlesungen iiber Zahlentheorie, I, 1901, 125-6. 
»«Proc. London Math. Soc, 34, 1901, 3-15. 
